International Journal of Game Theory

, Volume 37, Issue 3, pp 307–320 | Cite as

Symmetric versus asymmetric equilibria in symmetric supermodular games

  • Rabah AmirEmail author
  • Michał Jakubczyk
  • Małgorzata Knauff
Original Paper


This paper investigates the general properties of symmetric n-player supermodular games with complete-lattice action spaces. In particular, we examine the extent to which all pure strategy Nash equilibria tend to be symmetric for the general case of multi-dimensional strategy spaces. As asymmetric equilibria are possible even for strictly supermodular games, we investigate whether some symmetric equilibrium would always Pareto dominate all asymmetric equilibria. While this need not hold in general, we identify different sufficient conditions, each of which guarantees that such dominance holds: 2-player games with scalar action sets, uni-signed externalities, identical interests, and superjoin payoffs. Various illustrative examples are provided. Finally, some economic applications are discussed.


Strategic complementarity Endogenous heterogeneity Symmetry breaking Doubly symmetric games Superjoin payoffs 

JEL Classification

C72 L13 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Amir R (1996). Cournot oligopoly and the theory of supermodular games. Games Econ Behav 15: 132–148 CrossRefGoogle Scholar
  2. Amir R (2000). R&D returns, market structure and research joint ventures. J Inst Theor Econ 156: 583–598 Google Scholar
  3. Amir R and Lambson V (2000). On the effects of entry in Cournot markets. Rev Econ Stud 67: 235–254 CrossRefGoogle Scholar
  4. Amir R and Wooders J (2000). One-way spillovers, endogenous innovator/imitator roles and research joint ventures. Games Econ Behav 31: 1–25 CrossRefGoogle Scholar
  5. Athey S (2001). Single-crossing properties and existence of pure strategy equilibria in games of incomplete information. Econometrica 69: 861–890 CrossRefGoogle Scholar
  6. Bulow J, Geanokoplos J and Klemperer P (1985). Multimarket oligopoly: strategic substitutes and complements. J Polit Econ 93: 488–511 CrossRefGoogle Scholar
  7. Cooper R (1999). Coordination games: complementarities and macroeconomics. Cambridge University Press, Cambridge Google Scholar
  8. Echenique F (2003). The equilibrium set of two player games with complementarities is a sublattice. Econ Theory 22: 903–905 CrossRefGoogle Scholar
  9. Echenique F (2004). A characterization of strategic complementarities. Games Econ Behav 46: 325–347 CrossRefGoogle Scholar
  10. Echenique F (2005). A short and constructive proof of Tarski’s fixed-point theorem. Int J Game Theory 33: 215–218 CrossRefGoogle Scholar
  11. Fudenberg D and Tirole J (1984). The fat-cat effect, the puppy-dog ploy and the lean and hungry look. Am Econ Rev 74: 361–366 Google Scholar
  12. Hermalin B (1994). Heterogeneity in organizational form: why otherwise identical firms choose different incentives for their managers. Rand J Econ 25: 518–537 CrossRefGoogle Scholar
  13. Matsuyama K (2002). Explaining diversity: symmetry-breaking in complementarity games. Am Econ Rev 92: 241–246 CrossRefGoogle Scholar
  14. Milgrom P and Roberts J (1990). Rationalizability, learning and equilibrium in games with strategic complementarities. Econometrica 58(6): 1255–1277 CrossRefGoogle Scholar
  15. Milgrom P and Shannon C (1994). Monotone comparative statics. Econometrica 62: 157–180 CrossRefGoogle Scholar
  16. Mills D and Smith W (1996). It pays to be different: endogenous heterogeneity of firms in oligopoly. Int J Indus Organ 14: 317–329 CrossRefGoogle Scholar
  17. Monderer D and Shapley L (1996). Fictitious play property for games with identical interests. J Econ Theory 68: 258–265 CrossRefGoogle Scholar
  18. Shannon C (1994). Weak and strong monotone comparative statics. Econ Theory 5: 209–227 CrossRefGoogle Scholar
  19. Tarski A (1955). A lattice-theoretic fixed point theorem and its applications. Pacific J Math 5: 285–309 Google Scholar
  20. Topkis D (1978). Minimizing a submodular function on a lattice. Oper Res 26: 305–321 CrossRefGoogle Scholar
  21. Topkis D (1998). Supermodularity and complementarity. Princeton University Press, New Jersey Google Scholar
  22. Vives X (1990). Nash equilibrium with strategic complementarities. J Math Econ 19: 305–321 CrossRefGoogle Scholar
  23. Vives X (1999). Oligopoly pricing: old ideas and new tools. The MIT Press, Cambridge Google Scholar
  24. Weibull JW (1995). Evolutionary game theory. MIT Press, London Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Rabah Amir
    • 1
    Email author
  • Michał Jakubczyk
    • 2
    • 3
  • Małgorzata Knauff
    • 2
  1. 1.Department of EconomicsUniversity of ArizonaTucsonUSA
  2. 2.Warsaw School of EconomicsWarsawPoland
  3. 3.Department of PharmacoeconomicsMedical University of WarsawWarsawPoland

Personalised recommendations